Question: A $5$ inch by $7$ inch picture is placed in a frame that creates a uniform border of $x$ inches around the picture. $x$ $x$ The area of the entire frame (including where the picture is placed) is equal to $120$ square inches. Write an equation in terms of $x$ that models the situation.
The strategy We know that the area of the entire frame is $120$ square inches. Since the picture and frame are rectangular, we know that $120=l\cdot w$, where $l$ is the length of the frame and $w$ is the width of the frame. Now let's express $l$ and $w$ in terms of $x$. Expressing the length and width of the frame We know that the picture is $5$ inches wide and $7$ inches long. Since there is a uniform border of $x$ inches around the picture, we must add $2x$ inches to each dimension. Therefore, the width of the frame is $5+2x$ and the length of the frame is $7+2x$. $x$ $x$ $x$ $x$ $5+2x$ $7+2x$ Putting things together We found that $w=5+2x$ and $l=7+2x$. Since $l\cdot w=120$, we can substitute and find an equation in terms of $x$ that models the situation. The answer is: $ (7+2x)(5+2x)=120$